A Simple Approach to the 1:1 Resonance Bifurcation in Follower-Load Problems

We consider the problem of 1:1 resonance in autonomous, timereversible systems. We first present an abstract treatment for n-dimensionalsecond-order systems, and then apply our method to two simplemechanical examples involving follower loads. As the magnitude of the follower load is increased past a criticalvalue, the trivial solution loses stability as the real-valuedfrequencies of the linearized system first coalesce and then splitapart with complex-conjugate values. In Hamiltonian systems this isusually referred to as the Hamiltonian–Hopf bifurcation. Some novelfeatures of our analysis are the direct exploitation of reversibilityand a Liapunov–Schmidt reduction of the second-order (Newtonian)equations of motion, the latter of which requires no complexification.The analysis of the resulting two-parameter, one-dimensionalbifurcation equation yields the possibility that families ofnontrivial periodic solutions may exist for load values in excess of the critical value.